Knowledge of mathematics together with the ability to apply this knowledge to non-routine problems will be very valuable in the more and more automated world we face. Routine problems will tend to be solved mechanically, while new and challenging problems arise at a rapid rate for human minds to solve.

We hope that this book will be helpful in the process of mastering some aspects of mathematics and becoming proficient in using this knowledge. The topics selected contain important ideas that are often lost in the regular curriculum. An effort has been made to develop most of the theoretical material through sequences of related and progressively more sophisticated problems that follow the necessary text material and illustrative examples. In later chapters some proofs of a considerably more involved nature are omitted or dealt with only in special cases. The sets of problems provide opportunities for recognizing mathematical patterns and for conjecturing generalizations of specific results.

And now some words of advice to prospective problem solvers: some of the problems of this book are easy, some may take longer to solve than any previously encountered, and some may prove to be too difficult. If a problem is difficult, it may be helpful to look at the surrounding problems. One does not have to do all of the problems in a chapter before going on. In fact, some of the hard problems will appear to be easier if one returns to them after progressing through later chapters. The statements in problems preceded by the symbol "R" are required for subsequent work and should be specially noted.

Answers or hints are provided for the odd-numbered problems except starred problems and those that contain the answer in the problem. However, the greatest benefit comes from trying the problems; one should postpone looking at the answers as long as possible.